Question

Find the area and perimeter.

The figure shows a triangle. One side is given by 4 x minus 3 units. The second side is given by x plus 9 units. The third side is equal to 8 units. The height from the third side is given by 3 x units.

Hint: The height of a non-right triangle is the length of the segment of a line that is perpendicular to the base and that contains the opposite vertex

Answer 1

Answer:

Area=12x square units;  Perimeter=5x+14 units

Step-by-step explanation:

Recall the formulas for area and perimeter of a triangle:

Area of a triangle is half of the product of a base its corresponding height:

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

Perimeter is the sum of all of the side lengths:

$P=a+b+c$

In arbitrary triangles ABC, the sides are labeled across from a vertex with a matching lowercase letter.  So, in my diagram, side "b" is the bottom with length "8".  Side "b" is also the base, so the corresponding height is the perpendicular line segment that connects side "b" with vertex "B", which measures "3x"... 3 times some unknown value.

Side "a" is length "4x-3", and side "c" is length "x+9".

In this situation, where the sides have their specific relationship with "x" and the height is exactly triple "x", solving for x (and thus exact numerical values for the Perimeter and Area) involves trying to relate the two equations, possibly done with Heron's formula for area, or with trigonometry.  Ultimately, it involves solving a 4th degree polynomial to get an exact answer, which is not what the question intends, so we'll leave the Area and Perimeter in terms of x (meaning there will still be "x" in our expressions to represent the Area and Perimeter).

Area

$A=\frac{1}{2}bh\\A=\frac{1}{2}(8)(3x)\\A=12x$

So the Area of the triangle equals 12x square units.

Perimeter

$P=a+b+c\\P=(4x-3)+(8)+(x+9)\\P=5x+14$

So the Perimeter of the triangle equals 5x+14 units