Question

Hi, I am doing my geometry homework and am stuck on what methods to use to find the area. I tried drawing a bunch of heights but then the picture became so crammed I had to start over. Hints would be AWESOME!

Find the area of triangle ABC:

Answer 1

The area of triangle ABC is
`\( √(15) \)` square units.

To find the area of triangle ABC, we can use the fact that the perpendicular bisectors of the sides of a triangle meet at the circumcenter. The circumcenter is equidistant from the vertices of the triangle.

Given that AH = 4, HD = 1, and BD = DC, we can infer that the circumcenter (O) is also the centroid (since the median and perpendicular bisector coincide in an equilateral triangle). Therefore, AO = BO = CO.

Since HD = 1, AH = 4, and AO = BO, we can apply the Pythagorean theorem to find OD:

`\[ OD^2 = AO^2 - HD^2 \]`

`\[ OD^2 = 4^2 - 1^2 \]`

`\[ OD^2 = 16 - 1 \]`

`\[ OD^2 = 15 \]`

Since BD = DC and OD is the perpendicular bisector of BC, BD = DC =
`\( √(15) \)`. Now, we can find the area of triangle ABC using the area formula:

`\[ \text{Area} = (1)/(2) \cdot BC \cdot OD \]`

`\[ \text{Area} = (1)/(2) \cdot 2 \cdot √(15) \]`

`\[ \text{Area} = √(15) \]`

Therefore, the area of triangle ABC is
`\( √(15) \)` square units.