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# Geometry

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In triangle $ABC$, $AB = 10$ and $AC = 15$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 1:3$, then find $AD$.

Jun 2, 2024

#1
+806
+1

First off, let's let $$DC = 3BD$$

We can write two equations off the problem

$$AD = \sqrt{AC^2 - (3BD)^2 } = \sqrt{15^2 - 9BD^2 } \\ AD = \sqrt{AB^2 - BD^2} = \sqrt{10^2 - BD^2 }$$

Notice that we have AD on both equations, so we can equate the two to get

$$\sqrt{15^2 - 9BD^2} = \sqrt {10^2 -BD^2 }$$

Now, we just have to solve for BD^2.

$$15^2 - 9BD^2 = 10^2 - BD^2 \\ 225 - 9BD^2 = 100 - BD^2 \\ 125 = 8BD^2 \\ 125/8 = BD^2$$

We know that

$$AD =\sqrt{10^2 - BD^2}$$ so we can just plug in BD^2 to find AD.

$$AD = \sqrt{100 - 125/8} = \sqrt { 675/8} = \frac{15\sqrt3 }{2 \sqrt2}$$

Thanks! :)

Jun 2, 2024

#1
+806
+1

First off, let's let $$DC = 3BD$$

We can write two equations off the problem

$$AD = \sqrt{AC^2 - (3BD)^2 } = \sqrt{15^2 - 9BD^2 } \\ AD = \sqrt{AB^2 - BD^2} = \sqrt{10^2 - BD^2 }$$

Notice that we have AD on both equations, so we can equate the two to get

$$\sqrt{15^2 - 9BD^2} = \sqrt {10^2 -BD^2 }$$

Now, we just have to solve for BD^2.

$$15^2 - 9BD^2 = 10^2 - BD^2 \\ 225 - 9BD^2 = 100 - BD^2 \\ 125 = 8BD^2 \\ 125/8 = BD^2$$

We know that

$$AD =\sqrt{10^2 - BD^2}$$ so we can just plug in BD^2 to find AD.

$$AD = \sqrt{100 - 125/8} = \sqrt { 675/8} = \frac{15\sqrt3 }{2 \sqrt2}$$