Question

Given lJK , which is an isosceles right triangle, IJ=4 and m

Answer 1

`IK = IJ√(2)`

Explanation:

First, it's an isosceles right triangle, which means there is a 90 degree angle and the two other angles are equal.

To find what the two other angles are, you can set up a quick equation:

let x be angle

90 + 2x = 180, since the sum of the interior angles in a triangle is always 180.

then,

`2x = 90\\x = 45`

Therefore, the triangle is a 45-45-90 triangle.

You have the measure of IJ, which is one of the legs of the triangle. The way you can find the length of the hypotenuse of a 45-45-90 triangle is by multiplying the length of one of the legs by
`√(2)`.

This means the equation is
`IK = IJ√(2)`

Answer 2

`\sf IK=IJ√(2)`

Explanation:

The interior angles of a triangle sum to 180°. Therefore, if ΔIJK is an isosceles right triangle where m∠J = 90°:

• vertex is m∠J = 90°
• two base angles are m∠K and m∠I

To calculate the base angles:

⇒ m∠K + m∠I + m∠J = 180°

⇒ m∠K + m∠I + 90° = 180°

⇒ m∠K + m∠I = 90°

⇒ m∠K = m∠I = 45°

Therefore, IJ and JK are the legs of the right triangle and IK is the hypotenuse.

To find the length of the hypotenuse, use Pythagoras' Theorem.

Pythagoras’ Theorem:
`\sf a^2+b^2=c^2`

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

Given:

• a = IJ = 4
• b = JK = 4
• c = IK

Substitute the given values into the formula and solve for IK:

`\sf \implies IJ^2+JK^2=IK^2`

`\sf \implies 4^2+4^2=IK^2`

`\sf \implies IK^2=32`

`\implies \sf IK=√(32)`

`\implies \sf IK=√(16 \cdot 2)`

`\implies \sf IK=√(16)√(2)`

`\implies \sf IK=4√(2)`

As IJ = 4 then:

`\implies \sf IK=IJ√(2)`