Special Right Triangles Worksheet

Part A: Identify the Type of Special Right Triangle
1. Determine whether the following triangles are special right triangles. If yes, identify which type they are (45-45-90 or 30-60-90).
a) Triangle ABC with angles: ∠A = ∠B = 45°, ∠C = 90°
b) Triangle XYZ with angles: ∠X = ∠Y = 30°, ∠Z = 90°
c) Triangle DEF with angles: ∠D = 60°, ∠E = 30°, ∠F = 90°

2. For each of the following triangles, determine the missing side lengths using the appropriate ratios for special right triangles.
a) Triangle PQR is a 30-60-90 triangle with PQ = 6 cm. Find the lengths of QR and PR.
b) Triangle STU is a 45-45-90 triangle with ST = 8 inches. Find the lengths of SU and TU.
c) Triangle VWX is a 30-60-90 triangle with WX = 12 meters. Find the lengths of VW and VX.

3. The hypotenuse of a 45-45-90 triangle is 10 cm. Find the lengths of the legs.

4. The hypotenuse of a 30-60-90 triangle is 18 inches. Find the lengths of the legs.

Part B: Applications of Special Right Triangles
5. A ladder is leaning against a wall. The angle between the ladder and the ground is 60°. If the ladder is 12 feet long, how far is the base of the ladder from the wall?

6. An equilateral triangle is inscribed in a circle. What type of triangle is formed by connecting the center of the circle to two vertices of the equilateral triangle? Find the measure of each angle.

7. A ramp is being constructed to access a stage. If the height of the stage is 3 feet and the angle of inclination of the ramp is 30°, find the length of the ramp.

8. An architect is designing a triangular park. The park has a 90° corner and two sides that are 45 meters and 45 meters in length. What is the length of the diagonal of the triangular park?

9. A right triangle has legs of length 7 cm and 7√3 cm. Find the hypotenuse length.

10. A 30-60-90 triangle has a hypotenuse of length 20 units. Find the lengths of the legs.

Answers:
Part A:
1. a) Yes, 45-45-90 triangle.
b) Yes, 30-60-90 triangle.
c) Yes, 30-60-90 triangle.

2. a) QR = 6√3 cm, PR = 12 cm.
b) SU = 8 inches, TU = 8 inches.
c) VW = 6 meters, VX = 6√3 meters.
3. Leg lengths are both 5√2 cm.
4. Leg lengths are 9 inches and 9√3 inches.

Part B:
5. The base is 6√3 feet from the wall.
6. Isosceles triangle; each angle measures 60°.
7. The length of the ramp is 6 feet.
8. The diagonal length is 90 meters.
9. Hypotenuse length is 14 cm.
10. Leg lengths are 10 units and 10√3 units.

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