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.
AD ^ 2 = 9.
AD = 3.
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.
Answer: 10 cm.
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