Question

Write a paragraph proof: The sum of the lengths of the segments drawn from any point in the base of an isosceles triangle perpendicular to the legs is equal to the length of the altitude drawn to one leg.

Short answer

It is proved that,sum of the lengths of the segments drawn from any point in the base of an isosceles triangle perpendicular to the legs is equal to the length of the altitude drawn to one leg.

Step-by-step solution

Step 1. Draw a diagram with appropriate instruction.

CD perpendicular to AB. XY perpendicular to AC and XZ perpendicular to AB.

Step 2. Description of step.

By the Isosceles Triangle Theorem, angles $\angle ABC\cong \angle ACB$. The three triangles namely, $\Delta BZX,\Delta BDC$and $\Delta CYX$are similar. Then, $BX=CX,BZ=YC$and $ZX=YX$.

Hence, $\Delta BZX\cong \Delta CYX$.

Step 3. Description of step.

For similar triangles the ratio of sides is the same. Therefore,

$\frac{BX}{XZ}=\frac{CX}{XY}=\frac{BC}{CD}=k$

Step 4. Description of step.

As $BX+CX=BC$then,

$\begin{array}{c}kXZ+kXY=kCD\\ XZ+XY=CD\end{array}$

Hence proved.