Area of Triangle with Given Data

[alextrainer]

For b) area of AEF so one side is 7 - don't know how to get other 2 sides

not sure if right triangle; don't think so

how to use the data given since two of sides are slantedView attachment 6374

 

[MarkFL]

A triangle doesn't have to be a right triangle to use the area formula:

$$A=\frac{1}{2}bh$$

For $\triangle AEF$, we have $b=\overline{AF}=7$ and $h=\overline{AB}=5$

Thus, we get:

$$A=\frac{1}{2}\cdot7\cdot5=\frac{35}{2}$$

Suppose you aren't convinced we can use this formula except for right triangles. We could then drop a vertical line from point $E$ to $\overline{AD}$ and label the intersection $G$. $G$ will be to the right of $F$ and we'll say $\overline{FG}=x$

Now, we have the right triangle $AEG$, whose area is:

$$A_1=\frac{1}{2}(5+x)7$$

We also have the right triangle $FEG$ whose area is:

$$A_2=\frac{1}{2}(x)7$$

We can now find the area of $AEF$ by taking $A_1$, the area of the larger right triangle, and subtracting $A_2$, the area of the smaller right triangle:

$$A=A_1-A_2=\frac{1}{2}(5+x)7-\frac{1}{2}(x)7=\frac{1}{2}\cdot7\left((5+x)-x\right)=\frac{1}{2}\cdot7\cdot5=\frac{35}{2}$$

 

[alextrainer]

Thanks the line AB equals EF - what threw me off was EF looks at an angle - but I guess I assume since not drawn to scale - the 2 lines are the same.

 

[MarkFL]

Thanks the line AB equals EF - what threw me off was EF looks at an angle - but I guess I assume since not drawn to scale - the 2 lines are the same.

I would say, going by the diagram, that:

$$\overline{EF}>\overline{AB}$$

But, as I showed, we don't need to know $\overline{EF}$ since $\overline{AB}$ is the altitude of the triangle.

 

[phymat]

Thanks the line AB equals EF - what threw me off was EF looks at an angle - but I guess I assume since not drawn to scale - the 2 lines are the same.

They are not drawn to scale, but it doesn't mean that both are equal. Are you trying to find the area through herons formula? Or it is just that you were unaware about the simpler method?

 

[HallsofIvy]

You don't need to know the lengths of the sides of a triangle to find its area. The area of a triangle is (1/2)*base*height. Here the base is AF which we are told has length 7 and the height is 5.