# Triangle Problems

### Solutions

1Determine the area of an isosceles right triangle with the equal sides each measuring 10 cm in length.

2The perimeter of an equilateral triangle is 0.9 dm and its height is 25.95 cm. Calculate the area of the triangle.

3A 10 m long ladder is leaning against a wall. The bottom of the ladder is 6 m from where the wall meets the ground. At what height from the ground does the top of the ladder lean against the wall?

4Determine the side of an equilateral triangle whose perimeter is equal to a square of side 12 cm. Are their areas equal?

5Calculate the area of an equilateral triangle inscribed in a circle of radius 6 cm.

6The legs of a right triangle inscribed in a circle measure 22.2 cm and 29.6 cm. Calculate the circumference and the area of the circle.

7Given an equilateral triangle with a side of 6 cm, find the area of the circular sector determined by the circle circumscribed around the triangle and the radius passing through the vertices.

8The hypotenuse of a right triangle measures 405.6 m and the projection of a leg on it is 60 m in length. Calculate:

1 The length of the legs (catheti).

2 The height of the triangle.

3The area of the triangle.

9Calculate the sides of a triangle knowing that the projection of one of the legs on the hypotenuse is 6 cm and the height is cm.

## 1

Determine the area of an isosceles right triangle with the equal sides each measuring 10 cm in length.

A = (10 · 10) : 2 = 50 cm²

## 2

The perimeter of an equilateral triangle is 0.9 dm and its height is 25.95 cm. Calculate the area of the triangle.

P = 0.9 dm = 90 cm

l = 90 : 3 = 30 cm

A = (30 · 25.95) : 2 = 389.25 cm²

## 3

A 10 m long ladder is leaning against a wall. The bottom of the ladder is 6 m from where the wall meets the ground. At what height from the ground does the top of the ladder lean against the wall?

## 4

Determine the side of an equilateral triangle whose perimeter is equal to a square of side 12 cm. Are their areas equal?

P_{square} = 12 · 4 = 48 cm

P_{triangle} = 48 cml = 48 : 3 = 16 cm

A = 12^{2} = 144 m²

## 5

Calculate the area of an equilateral triangle inscribed in a circle of radius 6 cm.

The center of the circle is the centroid. Therefore:

## 6

The legs of a right triangle inscribed in a circle measure 22.2 cm and 29.6 cm. Calculate the circumference and the area of the circle.

## 7

Given an equilateral triangle with a side of 6 cm, find the area of the circular sector determined by the circle circumscribed around the triangle and the radius passing through the vertices.

The center of the circle is the centroid. Therefore:

## 8

The hypotenuse of a right triangle measures 405.6 m and the projection of a leg on it is 60 m in length. Calculate:

1 The length of the legs (catheti).

2 The height of the triangle.

3 The area of the triangle.

In all right triangles, the length one of the legs is a mean proportional between the hypotenuse and the projection on it.

In all right triangles, the height of the hypotenuse is a mean proportional between the two segments that it divides.

## 9

Calculate the sides of a triangle knowing that the projection of one of the legs on the hypotenuse is 6 cm and the height is cm.

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In all right triangles, the height of the hypotenuse is a mean proportional between the two segments that it divides.