Digital SAT Maths -Unit 4 - 4.6 Special Right Triangles- Study Notes- New Syllabus
Digital SAT Maths -Unit 4 – 4.6 Special Right Triangles- Study Notes- New syllabus
Digital SAT Maths -Unit 4 – 4.6 Special Right Triangles- Study Notes – per latest Syllabus.
Key Concepts:
30-60-90
45-45-90
Special Right Triangles — 30°-60°-90° Triangle
The DIGITAL SAT frequently uses special right triangles so you can find side lengths without using a calculator.
A 30°-60°-90° triangle has fixed side ratios.
Side Ratio
If the shortest side (opposite \( 30^\circ \)) is 1, then:
Shortest side = \( 1 \)
Longer leg (opposite \( 60^\circ \)) = \( \sqrt{3} \)
Hypotenuse = \( 2 \)
So the ratio is:
\( 1 : \sqrt{3} : 2 \)
Important Identification Rule
- Shortest side is always opposite \( 30^\circ \)
- Longest side is always the hypotenuse
Scaling Rule
If the smallest side is \( k \), then:
Longer leg = \( k\sqrt{3} \)
Hypotenuse = \( 2k \)
DIGITAL SAT Tip
If you see \( 30^\circ \), \( 60^\circ \), or a \( \sqrt{3} \) in a right triangle problem, it is almost certainly a 30-60-90 triangle.
Example 1 (Find Hypotenuse):
In a 30°-60°-90° triangle, the shortest side is 5. Find the hypotenuse.
▶️ Answer/Explanation
Hypotenuse \( = 2k = 2(5) = 10 \)
Answer: 10
Example 2 (Find Longer Leg):
The hypotenuse of a 30°-60°-90° triangle is 14. Find the longer leg.
▶️ Answer/Explanation
\( 2k = 14 \Rightarrow k = 7 \)
Longer leg \( = k\sqrt{3} = 7\sqrt{3} \)
Answer: \( 7\sqrt{3} \)
Example 3 (SAT Style):
A ladder leans against a wall making a \( 60^\circ \) angle with the ground. The base of the ladder is 6 ft from the wall. How high up the wall does the ladder reach?
▶️ Answer/Explanation
The ground distance is the shorter leg (opposite \( 30^\circ \)).
\( k = 6 \)
Height \( = k\sqrt{3} = 6\sqrt{3} \)
Answer: \( 6\sqrt{3} \) ft
Special Right Triangles — 45°-45°-90° Triangle
A 45°-45°-90° triangle is an isosceles right triangle. This means the two legs are equal because the two acute angles are equal.
Side Ratio
If each leg is 1, then the hypotenuse is:
\( \sqrt{1^2 + 1^2} = \sqrt{2} \)
So the ratio is:
\( 1 : 1 : \sqrt{2} \)
Scaling Rule
If each leg is \( k \):
Hypotenuse \( = k\sqrt{2} \)
Important Identification Clues
- A right triangle with equal legs
- A square cut along the diagonal
- Presence of \( 45^\circ \)
DIGITAL SAT Tip
Whenever you see a square’s diagonal or a \( 45^\circ \) angle, immediately think 45-45-90 triangle.
Example 1 (Find Hypotenuse):
A right triangle has legs of length 8 and 8. Find the hypotenuse.
▶️ Answer/Explanation
Hypotenuse \( = 8\sqrt{2} \)
Answer: \( 8\sqrt{2} \)
Example 2 (Find a Leg):
The hypotenuse of a 45°-45°-90° triangle is \( 10\sqrt{2} \). Find each leg.
▶️ Answer/Explanation
\( k\sqrt{2} = 10\sqrt{2} \)
\( k = 10 \)
Answer: each leg = 10
Example 3 (Square Diagonal SAT Style):
A square has side length 6. Find the length of its diagonal.
▶️ Answer/Explanation
The diagonal forms a 45-45-90 triangle.
Diagonal \( = 6\sqrt{2} \)
Answer: \( 6\sqrt{2} \)