## SAT MAth Practice questions – all topics

**Geometry and Trigonometry**Weightage: 15% Questions: 5-7- Area and volume
- Lines, angles, and triangles
- Right triangles and trigonometry
- Circles

## SAT MAth and English – full syllabus practice tests

**[Calc]**** ****Question** ** **** Hard**

Line \(m\) is shown in the \(x y\)-plane, and the point with coordinates \((0.25, r)\) is on line \(m\). What is the value of \(r\) ?

**▶️Answer/Explanation**

**Ans:5/3**

To find the equation of line \(m\), we can first determine its slope using the given points \((0,2)\) and \((1.5,0)\). Then, we can use the slope-intercept form of a line (\(y = mx + b\)), where \(m\) is the slope and \(b\) is the y-intercept.

Calculate the slope (\(m\)):

\[m = \frac{{y_2 – y_1}}{{x_2 – x_1}}\]

\[m = \frac{{0 – 2}}{{1.5 – 0}}\]

\[m = \frac{{-2}}{{1.5}}\]

\[m = -\frac{4}{3}\]

Use the slope-intercept form to find the equation of the line:

Given that the y-intercept is 2, we can substitute \(m\) and \(b\) into the equation:

\[y = -\frac{4}{3}x + 2\]

So, the equation of line \(m\) is \(y = -\frac{4}{3}x + 2\).

Now, we’re given that the point \((0.25, r)\) is on line \(m\). We can substitute \(x = 0.25\) into the equation of line \(m\) to find \(r\):

\[r = -\frac{4}{3}(0.25) + 2\]

\[r = -\frac{1}{3} + 2\]

\[r = \frac{5}{3}\]

Therefore, the coordinates of the point on line \(m\) are \((0.25, \frac{5}{3})\).

**[Calc]**** ****Question** ** **** Hard**

For the linear equation \(y=m x+b\), where \(m\) and \(b\) are positive constants, which of the following tables gives three values of \(x\) and their corresponding values of \(y\) ?

**▶️Answer/Explanation**

**Ans:B**

Given the linear equation \( y = mx + b \), we need to match the given values of \( x \) with their corresponding values of \( y \) from the options.

1. For \( x = -2 \):

\[

y = m(-2) + b = -2m + b

\]

2. For \( x = 1 \):

\[

y = m(1) + b = m + b

\]

3. For \( x = \frac{-b}{m} \):

\[

y = m \left( \frac{-b}{m} \right) + b = -b + b = 0

\]

**Option A:**

For \( x = -2 \): \( y = -2m \) (incorrect, should be \( -2m + b \))

For \( x = 1 \): \( y = m \) (incorrect, should be \( m + b \))

For \( x = \frac{b}{m} \): \( y = 0 \) (incorrect \( x \) value)

** Option B:**

For \( x = -2 \): \( y = -2m + b \) (correct)

For \( x = 1 \): \( y = m + b \) (correct)

For \( x = \frac{-b}{m} \): \( y = 0 \) (correct)

** Option C:**

For \( x = -2 \): \( y = -2m + b \) (correct)

For \( x = 1 \): \( y = m + b \) (correct)

For \( x = b \): \( y = 0 \) (incorrect \( x \) value)

** Option D:**

For \( x = -2 \): \( y = -2m \) (incorrect, should be \( -2m + b \))

For \( x = 1 \): \( y = m \) (incorrect, should be \( m + b \))

For \( x = b \): \( y = 0 \) (incorrect \( x \) value)

**[No calc]**** ****Question** ** **Hard

Triangle ABC and triangle DEF each have two angles measuring 35°, as shown. Which of the following additional pieces of information is sufficient to prove that triangle ABC is congruent to triangle DEF ?

A) the measures of LACB and ∠DFE are equal

B) The lengths of \(\overline{BC}\) and \(\overline{EF}\) are equal

C) The lengths of AC and DE are equal.

D)No additional information is necessary to prove that the two triangles are congruent.

**▶️Answer/Explanation**

**B) The lengths of \(\overline{BC}\) and \(\overline{EF}\) are equal**

To prove that two triangles are congruent, we need to satisfy one of the triangle congruence conditions. Given the information provided in the diagram, we can use the Angle-Side-Angle (ASA) congruence condition.

The ASA congruence condition states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

In this case, we have: ∠A = 35° and ∠B = 35° ∠D = 35° and ∠E = 35°

So we know that two angles of triangle ABC are congruent to two angles of triangle DEF.

The additional piece of information needed is to ensure that the included side between those congruent angles is also congruent.

The lengths of \(\overline{BC}\) and \(\overline{EF}\) are equal

Therefore, option B is sufficient to prove that triangle ABC is congruent to triangle DEF by the ASA congruence condition.

**[Calc]**** ****Question** ** Hard**

The measure of angle A is \(\frac{7}{12}\pi\) radians greater than the measure of angle B. How much greater is the measure of angle A than the measure of angle B, in degrees ? (Disregard the degree symbol when entering your answer.)

**▶️Answer/Explanation**

105

We need to find how much greater the measure of angle \(A\) is than the measure of angle \(B\), in degrees, given that \(\text{angle } A\) is \(\frac{7}{12}\pi\) radians greater than \(\text{angle } B\).

1. Convert \(\frac{7}{12}\pi\) radians to degrees:

Use the conversion factor \(\frac{180^\circ}{\pi}\):

\[

\frac{7}{12}\pi \times \frac{180^\circ}{\pi} = \frac{7}{12} \times 180

\]

\[

= \frac{7 \times 180}{12} = \frac{1260}{12} = 105^\circ

\]

Thus, the measure of angle \(A\) is \(105\) degrees greater than the measure of angle \(B\).

**[Calc]**** ****Question** ** ****Hard**

Triangle LMN and triangle PQR each have an angle measuring 60° and a given side length, as shown.

For triangles LMN and PQR, which additional piece of information is sufficient to prove that the triangles are similar?

I. The length of line segment PQ is \(\frac{2}{3}\) the length of line segment LM.

II. The length of line segment PR is \(\frac{2}{3}\) the length of line segment LN.

A) I is sufficient but II is not.

B) II is sufficient but I is not.

**▶️Answer/Explanation**

C) I is sufficient and II is sufficient.

D) Neither I nor II is sufficient.

C) I is sufficient and II is sufficient.

For these triangles to be similar

\[

\begin{aligned}

\frac{M N}{Q R} & =\frac{L N}{P R}=\frac{L M}{P Q} \\

\frac{9}{6} & =\frac{L N}{P R}=\frac{L M}{P Q} \\

\frac{3}{2} & =\frac{L N}{P R}=\frac{L M}{P Q} \\

\frac{P R}{L N} & =\frac{P Q}{L M}=\frac{2}{3} .

\end{aligned}

\]

**[Calc]**** ****Question** ** ****Hard**

Triangle KLM (shown) is similar to triangle RST (not shown). For these triangles ∠R and ∠S correspond to ∠K and ∠L respectively and RS = 3KL. Which

of the following statements is(are) true ?

I. The measure of LR is 45°.

II. ST= 60

A) I only

B) II only

C) I and II

D) Neither I nor II

**▶️Answer/Explanation**

B) II only

- Statement I claims “The measure of \(\angle L R\) is \(45^{\circ}\).” To analyze this, we need to understand the context of \(\angle L R\). However, \(\angle L R\) seems to be a typo or misinterpretation since \(\angle L\) in the context of triangle KLM or RST should be considered.
- Since \(\angle L\) in triangle KLM corresponds to \(\angle S\) in triangle RST and we don’t have the value of \(\angle L\) directly, we cannot conclude \(\angle L R=45^{\circ}\) based on the given data. This statement appears to be incorrect or misleading due to possible mislabeling.
- Statement II claims “ST \(=60\).”
- If \(R S=3 K L\), then the corresponding sides in triangle RST are 3 times those in triangle KLM. Given \(L M=20\) in triangle KLM, the corresponding side \(S T\) in triangle RST should be \(3 \times 20=60\).

Therefore, the correct answer is: B) II only

**[Calc]**** ****Question**** **** Hard**

Trapezoid A and trapezoid B shown are similar. The length of each side of trapezoid $A$ is 8 times the length of the corresponding side of trapezoid $\mathrm{B}$. The area of trapezoid A is how many times as large as the area of trapezoid B?

A) 8

B) 16

C) 32

D) 64

**▶️Answer/Explanation**

D

**[Calc]**** ****Question**** **** Hard**

The graph of $y=m x+b$, where $m$ and $b$ are constants, is shown in the $x y$-plane.

What is the value of $m$ ?

**▶️Answer/Explanation**

2

*Question*

In the \(xy\)-plane, line \(k\) with equation \(y = mx + b\), where \(m\) and \(b\) are constants, passes through the point (-3, 1). If line \(k\) is perpendicular to the line with equation \(y = – 2x + 3\), what is the value of \(b\) ? 1.6

**▶️Answer/Explanation**

5/2, 2.5

*Question*

.

The scatterplot shows 12 values from a data set. A line of best fit for the data is also shown. Which of the following is the best interpretation of the \(y\)-coordinate of the \(y\)-intercept of the line of best fit? 3.5

- For the value \(x\) =6, the line of best fit predicts the corresponding \(y\)-value to be approximately 0.
- For the value \(y\) =0, the line of best fit predicts the corresponding \(x\)-value to be approximately 3.
- For the value \(x\) = 0 , the line of best fit predicts the corresponding \(y\)-value to be approximately 6.
- For the value \(y\) = 3 , the line of best fit predicts the corresponding \(x\)-value to be approximately 0.

**▶️Answer/Explanation**

C

*Question*

It took 20 minutes for a jet to climb from a starting altitude of 10,000 feet to a final altitude of 30,000 feet. If the jet climbed at a constant rate, what was its altitude, in feet, 14 minutes after the climb began?

- 14,000
- 21,000
- 24,000
- 28,000

**▶️Answer/Explanation**

C

*Question*

In right triangle , . What is the measure, in degrees, of angle X?

**▶️Answer/Explanation**

70

*Questions *

$a x+b y=b$

In the equation above, $\mathrm{a}$ and $\mathrm{b}$ are constants and $o<a<b$. Which of the following could represent the graph of the equation in the $x y$-plane?

**▶️Answer/Explanation**Ans: C

*Questions *

In the $x y$-plane, line $k$ intersects the $y$-axis at the point $(0,-6)$ and passes through the point $(2,2)$. If the point $(20, w)$ lies on line $k$, what is the value of $w$ ?

**▶️Answer/Explanation**

Ans: 74

* *

*Question*

In the $x y$-plane above, line $k$ passes through the points $(0,2)$ and $(3,0)$. If the line $k$ is defined by the equation $y=m x+b$, where $m$ and $b$ are constants, what is the value of $b$ ?

**▶️Answer/Explanation**

Ans: 2

*Questions *

$4 x-5 y=2$

The graph of the equation above in the $x y$-plane is a line. What is the $x$-coordinate of the $x$-intercept of the line?

**▶️Answer/Explanation**

Ans: 1/2, . 5

*Question*

In the $x y$-plane, line $k$ passes through the point $(3,1)$ and is parallel to the line with equation $y=\frac{5}{2} x-\frac{7}{2}$. What is the slope of line $k$ ?

**▶️Answer/Explanation**

Ans: $5 / 2,2.5$

*Question*

The two acute angles of a right triangle have degree measures of $x$ and $y$. If $\sin x=\frac{5}{13}$, what is the value of $\cos y$ ?

**▶️Answer/Explanation**

Ans: 5/13, .384, . 385

*Question*

Lines $t$ and $w$ are parallel in the $x y$-plane. The equation of line $t$ is $4 x+7 y=14$, and line $w$ passes through $(-3,8)$. What is the value of the $y$ intercept of line $w$ ?

**▶️Answer/Explanation**

Ans: 44/7,6.28, 6.29

*Questions *

An angle measure of 540 degrees was written in radians as $x \pi$. What is the value of $x$ ?

**▶️Answer/Explanation**

Ans: 3

*Question*

An angle with a measure of $\frac{7 \pi}{6}$ radians has a measure of $d$ degrees, where $0 \leq d<360$. What is the value of $d$ ?

**▶️Answer/Explanation**

Ans: 210

*Question*

A line is shown in the \(xy\)-plane above. A second line (not shown) is parallel to the line shown and passes through the points (1, 1) and \((3, c)\), where \(c\) is a constant. What is the value of \(c\) ?

**▶️Answer/Explanation**

Ans: 2.5, 5/2