Question

The height of a triangle is 5 cm shorter than its base. If the area of the triangle is 25 cm2, find the height of the triangle.

Answer 1

Answer:

$h = 5\ cm$

Step-by-step explanation:

Let's call B at the base of the triangle and call h at the height of the triangle. Then we know that:

The height of a triangle is 5 cm shorter than its base. This means that:

$h = B-5$.

 The area of the triangle is 25 cm²

By definition the area of a triangle is:

$A = 0.5Bh$

For this triangle we know that $A = 25\ cm^2$ and $h = B-5$. We substitute these values in the equation and solve for B.

$25 = 0.5B (B-5)$

$0.5B ^ 2-\frac{5}{2}B-25 = 0$

Now we use the quadratic formula to solve the equation.

For an equation of the form $ax ^ 2 + bx + c = 0$ the quadratic formula is:

$B=\frac{-b\±\sqrt{b^2-4ac}}{2a}$

In this case note that:  $a=0.5,\ \ b=-\frac{5}{2}\ \ c=-25$

Then:

$B=\frac{-(-\frac{5}{2})\±\sqrt{(-\frac{5}{2})^2-4(0.5)(-25)}}{2(0.5)}$

$B=\frac{\frac{5}{2}\±\sqrt{\frac{25}{4}+50}}{1}$

$B=\frac{5}{2}\±\frac{15}{2}$

The solutions are:

$B_1=\frac{5}{2}+\frac{15}{2}=10$

$B_2=\frac{5}{2}-\frac{15}{2}=-5$

For this problem we take the positive solution.

$B=10\ cm$

Now we substitute the value of B in the equation to find the height h

$h = 10-5$

$h = 5\ cm$

Answer 2

The height of the triangle is: 5 cm.

Area of a Rectangle

Area of a rectangle is given as, A = 1/2(base)(height).

Given:

• Base = x = ?
• Height (h) = (x - 5)
• Area (A) = 25 sq. cm

Substitute the values

1/2(x)(x - 5) = 25

x² - 5x = 50

x² - 5x - 50 = 0

• Factorize

(x + 5)(x - 10)

x = -5 or x = 10

The base would be the positive value of x.

• Therefore:

Height (h) = (x - 5)

Plug in the value of x

Height (h) = 10 - 5

Height (h) = 5 cm

Therefore, the height of the triangle is: 5 cm.

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