Question

Two points D and E are taken inside the triangle ABC such as <ABD = <EBC
From the point D two perpendicular lines DF and DG are drawn to AB and BC
respectively . From the point E two perpendicular lines EH and EI are drawn to
AB and BC respectively . DF=7,DG=8,EI=15, EH=?

Answer 1

Answer on Question #72127 - Math - Geometry

Question 72127:

Two points $D$ and $E$ are taken inside the triangle ABC such as $\angle ABD = \angle EBC$.

From the point $D$ two perpendicular lines $DF$ and $DG$ are drawn to $AB$ and $BC$ respectively.

From the point $E$ two perpendicular lines $EH$ and $EI$ are drawn to $AB$ and $BC$ respectively.

$DF = 7, DG = 8, EI = 15, EH = ?$

Answer:

Let $E'$ be a point on the line $BD$ with perpendicular lines $E'H'$ and $E'I'$ drawn to $AB$ and $BC$ respectively, such that:

$$E'H' = EI \text{ and } E'I' = EH$$

The second equality results from the facts that $\Delta E'H'B = \Delta EIB$ and $\Delta E'I'B = \Delta EHB$.

Since $\Delta E'H'B \approx \Delta DFB$ and $\Delta E'I'B \approx \Delta DGB$, we have that:

$$EH = E'I' = DG \cdot E'H'/DF = DG \cdot EI/DF = 8 \cdot 15/7 = 17.14.$$

Answer:

$EH = 17.14$.