Math CentralQuandaries & Queries


Question from Bailey, a student:

The length of the base of a triangle is 1cm less than 5 times the height of the triangle. The area of the triangle is 21 square cm. What is the the height of the triangle and the length of the base of the triangle?

Hi Bailey.

The area of a triangle is (1/2) times the base times the height.

Let b = the base length, h = the height, and A = the area. Then

A = (1/2) b h.

You said "the length of the 1cm less than 5 times the height...". So:

b = 5h - 1.

Therefore you can substitute this expression for b into the area equation. Thus A = (1/2) b h becomes:

A = (1/2) (5h - 1) h.

You said "the 21 square cm". So A = 21.

21 = (1/2) (5h - 1) h.

This is a quadratic equation. When we put it in standard form we get:

0 = 5h2 - h - 42

Now can you solve for the height (h) using factoring or completion of the square? Remember that the height of a triangle must be a positive number. Then you can use that to find b using b = 5h - 1.

Write us back if you are still stuck (tell us where you have trouble).

Stephen La Rocque

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