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
Question Medium
In the figure shown, points \(A\) and \(B\) lie on the circle with radius 1 centered at the origin, \(O\). If the cosine of \(\angle A O B\) is \(-\frac{1}{2}\), what is the measure, in radians, of \(\angle A O B\) ?
A) \(\frac{5 \pi}{12}\)
B) \(\frac{7 \pi}{12}\)
C) \(\frac{2 \pi}{3}\)
D) \(\frac{5 \pi}{6}\)
▶️Answer/Explanation
Ans:C
\(cos~ \angle \rm{AOB}=-\frac{1}{2}\)
$\rm cos~ \angle\rm{AOB}=\frac{\rm{OA}^2+\rm{OB}^2 – \rm{AB}^2}{2\times\rm{OA}\times \rm{OB}}$
$\frac{-1}{2}=\frac{\rm{1}^2+\rm{1}^2 – \rm{AB}^2}{2\times\rm{1}\times \rm{1}}$
$-1=2-\rm{AB}^2$
$3=\rm{AB}^2$
$\rm{AB}=\sqrt 3$
We are given \(\cos (\angle A O B)=-\frac{1}{2}\).
Recall that \(\cos (\theta)=-\frac{1}{2}\) for angles \(\theta\) in the range from 0 to \(2 \pi\) (or 0 to 360 degrees).
The standard angles where \(\cos (\theta)=-\frac{1}{2}\) are:
\(120^{\circ}\) (which is \(\frac{2 \pi}{3}\) radians)
\(240^{\circ}\) (which is \(\frac{4 \pi}{3}\) radians)
Question Medium
The function \(f\) is defined by \(f(x)=2 x+6\). What is the graph of \(y=f(x)\) ?
▶️Answer/Explanation
A
Y-intercept
The \(y\)-intercept occurs where the graph of the function crosses the \(y\)-axis. This happens when \(x=0\).
\[
f(0)=2(0)+6=6
\]
Thus, the \(y\)-intercept is at \((0,6)\).
X-intercept
The \(\mathrm{x}\)-intercept occurs where the graph of the function crosses the \(\mathrm{x}\)-axis. This happens when \(f(x)=0\).
\[
2 x+6=0
\]
Solve for \(x\) :
\[
\begin{aligned}
& 2 x=-6 \\
& x=-3
\end{aligned}
\]
Thus, the \(x\)-intercept is at \((-3,0)\).
Question medium
Which of the following pieces of information is sufficient to prove that triangle ABC is an isosceles triangle?
- \(\overline{AB}\) is congruent to \(\overline{BC}\)
- ∠A is congruent to ∠C
A) I is sufficient, but II is not
B) II is sufficient, but I is not
C) Either I or II is sufficient
D) Neither I nor II is sufficient
▶️Answer/Explanation
C) Either I or II is sufficient
To determine which piece of information is sufficient to prove that triangle \(ABC\) is an isosceles triangle, let’s analyze each statement:
Statement I: \(\overline{AB}\) is congruent to \(\overline{BC}\)
If \(\overline{AB}\) is congruent to \(\overline{BC}\), then by definition, triangle \(ABC\) has two sides that are equal in length. This is sufficient to prove that triangle \(ABC\) is an isosceles triangle.
Statement II: \(\angle A\) is congruent to \(\angle C\)
If \(\angle A\) is congruent to \(\angle C\), this implies that the angles opposite those angles (\(\overline{BC}\) and \(\overline{AB}\) respectively) are equal. According to the Isosceles Triangle Theorem, if two angles in a triangle are equal, then the sides opposite those angles are also equal. Therefore, this is sufficient to prove that triangle \(ABC\) is an isosceles triangle.
Both statements independently provide sufficient information to prove that triangle \(ABC\) is an isosceles triangle.
Question Medium
Triangles \(\mathrm{ABC}\) and \(\mathrm{DEF}\) each have a corresponding angle measuring \(40^{\circ}\). Which additional piece of information is sufficient to determine whether these two triangles are similar?
A) The length of line segment \(A C\)
B) The length of line segment \(\mathrm{DE}\)
C) The measure of another pair of corresponding angles in the two triangles.
D) The lengths of one pair of corresponding sides in the two triangles.
▶️Answer/Explanation
Ans:C
To determine if two triangles are similar, we need to check if their corresponding angles are equal or if their corresponding sides are proportional.
Given that triangles ABC and DEF each have a corresponding angle measuring \(40^\circ\), the additional piece of information needed to determine similarity is:
C) The measure of another pair of corresponding angles in the two triangles.
By knowing two corresponding angles, we can use the Angle-Angle (AA) similarity criterion. If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
The answer is:
\[
\boxed{\text{C}}
\]
Question Medium
In the figure shown, \(\bar{AE}\) and \(\bar{BD}\) intersect at point C. Which of the following additional pieces of information is NOT sufficient to prove that \(\bigtriangleup \)ABC is similar to \(\bigtriangleup \)EDC ?
A. \(\bar{AB}\) is parallel to \(\bar{DE}\) .
B. The measure of \(\angle D\) is equal to the measure of \(\angle B\).
C. The length of \(\bar{AB}\)is equal to the length of \(\bar{DE}\).
D. The measure of \(\angle A\) is equal to the measure of \(\angle B\), and the measure of \(\angle D\)is equal to the measure of \(\angle E\).
▶️Answer/Explanation
Ans: C
The length of \(\bar{AB}\) is equal to the length of \(\bar{DE}\).”
If the lengths of two corresponding sides in two triangles are equal, it implies that all the corresponding sides are proportional with a ratio of 1:1. However, this alone does not guarantee that the corresponding angles are equal, which is another necessary condition for proving similarity between triangles.