# GMAT Quant: Geometry – Special Right Triangles

Questions on Special Right Triangles, Pythagorean theorem or triplets are the most common types of questions asked in GMAT Quant. Triangles are the most basic geometrical figures that one learns in geometry. But, even these simplest geometrical forms have some varieties.

Special right triangle is a triangle that has at least one of its angles equal 90°The side which is opposite to the right angle is known as the hypotenuse of the triangle; whereas, the rest of the sides are known as the legs. There is no special name provided for the other two angles, but the sum of these two angle is always complementary. The reason of holding this property is; the sum of all the three angles of a triangle is 180 degrees, and since one of the angle is the right angle, so the sum of the other angles will be 90 degrees.

The special right triangles can be investigated on four platforms:

• 45°-45°-90° triangle
• 30°-60°-90° triangle
• 5-12-13 triangles
• 3-4-5 triangle

Let’s discuss the properties of these triangles individually.

 Type Properties Figure 45°-45°-90° It is a right angle triangle whose two sides and two angles are equal. So, it is an isosceles triangle as well. The length of the sides follow in a ratio of $1 : 1 : \sqrt{2}$ $Hypotenuse = Leg \times \sqrt{2}$ $Leg = \frac{Hypotenuse}{\sqrt2}$ 30°-60°-90° The Hypotenuse is always opposite to the right angle. The shorter leg is opposite to the 30°. The length of the sides of the triangle are in a ratio of 1:√3:2 $The \; Shorter \; Side = \frac{Hypotenuse}{2}$ Long length =2×short side Hypotenuse=√3×short side 5-12-13 It is a right-angle triangle where the lengths are in a ratio of 5:12:13. It follows Pythagorean Theorem as well. It is a common triplet. 3-4-5 The legs are in a ratio of 3 : 4 : 5

Let’s solve some question and understand our strengths and weaknesses in this topic:

Question: What is the area of the right triangle ABC?

Solution: In order to find the area, we need a height and a base. If only we knew the length of AB, then AB will be the height and AC can be taken as the base, since these two sides are perpendicular to one another.

Pythagorean Theorem can be used to find the length of side AB.

So, according to the theorem, (AB) 2 + (AC) 2 = (BC) 2

(AB) 2 + (12)2 = (13)2

(AB) 2 + 144 = 169

(AB) 2 = 25

AB = 5

So, area =1/2×5×12=30 sq. units

Question: In the figure shown below, find the value of x?

1. 2.5
2. 5/√2
3. 10
4. 5√2
5. 10/√2

Solution: Let’s redraw the diagram, labelling all the angles, using the axiom that sum of all the angles of a triangle is equal to 180°.

Now, it can be clearly seen that it is forming a two different 45°- 45°- 90° triangle. Hence, their sides will be in a ratio of 1:1: √2. So, the length of each leg will be equal to $\frac{Hypotenuse}{\sqrt2}$.

Since the length of hypotenuse of the larger triangle is equal to 20, so its sides will be $\frac{20}{\sqrt2} = 10 \sqrt2$

Adding more labels to the figure, it will become

Since, the hypotenuse of the smaller triangle is known. So, again apply the same axiom of 1:1: √2. Hence, x = 10.