# At the base of the straight prism lies an isosceles triangle ABC, AB = BC = 5 cm. The height BD of triangle ABC is 4 cm.

At the base of the straight prism lies an isosceles triangle ABC, AB = BC = 5 cm. The height BD of triangle ABC is 4 cm. Find the length of the diagonal of the face of the prism containing the base of the triangle if the height of the prism is 8 cm.

In order to find the diagonal of the prism face, we need to find all the dimensions of the prism face – the lower length and height of the prism.

Find the bottom length:

The lower length of the prism is equal to the base of the isosceles triangle – the AC side.

Consider triangles ABD and DBC:

Since the two sides of a given triangle are equal, these triangles are equal:

Find the side AD (DC) by the Pythagorean theorem:

AB (BC) – hypotenuse.

AD, BD (DC, BD) – leg.

5 ^ 2 = 4 ^ 2 + AD ^ 2.

AD ^ 2 = 5 ^ 2 – 4 ^ 2

AD ^ 2 = 25 – 16 = 9.

The base of the triangle ABC = AC = 2 * AD = 2 * 3 = 6.

The base is 6.

Consider a right-angled triangle: AC, height, diagonal of the prism.

AC – 6 cm, height – 8 cm.

d ^ 2 = 8 ^ 2 + 6 ^ 2 = 64 + 36 = 100.

d = 10 cm. 