Area of Isosceles Triangle proof

In summary, to arrive at the third line of this proof, the expression $h(p+q)$ is multiplied by $ab/ab$ and then simplified using the distributive property and the fact that $ab/ab=1$. This results in the final expression of $\frac{1}{2}ab\left(\frac{p}{a}\frac{h}{b} + \frac{h}{a}\frac{q}{b}\right)$ on the third line. If you need further clarification, please let me know. Thanks!
  • #1
paulmdrdo1
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I want to know how did it arrive at the third line of this proof. I didn't get it. Thanks!
 

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  • #2
paulmdrdo said:
I want to know how did it arrive at the third line of this proof. I didn't get it. Thanks!

Hi paulmdrdo,

First multiply $h(p + q)$ by $ab/ab$ (which equals $1$) to get the expression

\(\displaystyle ab \cdot \frac{h(p + q)}{ab}.\)

Since

\(\displaystyle \frac{h(p + q)}{ab} = \frac{hp+hq}{ab} = \frac{p}{a}\frac{h}{b} + \frac{h}{a}\frac{q}{b},\)

we have

\(\displaystyle ab \cdot \frac{h(p + q)}{ab} = ab\left(\frac{p}{a}\frac{h}{b} + \frac{h}{a}\frac{q}{b}\right)\)

and therefore

\(\displaystyle \frac{1}{2}h(p + q) = \frac{1}{2}ab\left(\frac{p}{a}\frac{h}{b} + \frac{h}{a}\frac{q}{b}\right).\)
 

FAQ: Area of Isosceles Triangle proof

What is an isosceles triangle?

An isosceles triangle is a type of triangle that has two sides of equal length. This means that two of its three angles are also equal. The third angle, called the base angle, is different from the other two.

How do you find the area of an isosceles triangle?

The area of an isosceles triangle can be found by multiplying the base length (b) by the height (h) and dividing the result by 2. The formula for the area of an isosceles triangle is A = (b x h) / 2.

Why is the area of an isosceles triangle proof important?

The area of an isosceles triangle proof is important because it helps us understand and prove the relationship between the sides and angles of an isosceles triangle. It also allows us to find the area of an isosceles triangle without measuring all three sides.

How do you prove the area of an isosceles triangle?

There are several methods for proving the area of an isosceles triangle, including using the Pythagorean theorem, the Law of Sines, and the Law of Cosines. One common method is to split the isosceles triangle into two right triangles, and then use the formula for the area of a triangle (A = 1/2 x base x height) to find the area of each right triangle. The sum of these two areas is equal to the area of the isosceles triangle.

How do you apply the area of an isosceles triangle proof in real life?

The area of an isosceles triangle proof has practical applications in fields such as engineering, architecture, and construction. It can be used to calculate the area of a roof, the amount of material needed for a triangular-shaped structure, or the area of a piece of land with an isosceles triangle shape. It also helps us understand and solve problems involving isosceles triangles in various situations.

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