Test 4 Review

Areas

100

True or False: If two triangles have the same areas and the same length of altitude, these triangles must be congruent.

What is False, They can have the same base and altitude, but they can be different shapes.
Congruent triangles are a subset of triangles that have the same area.

200

Let R and S be two enclosed regions. What additional constraint must I have for these regions to be able to state that
A_R + A_S = A_R+S

What is the regions must not overlap.
If they overlap, then the sum of the areas of the two regions does not equal the sum of each of the individual regions.

300

What is the equation for the area of a rectangle whose base has length b and height has length h?

What is A = bh

400

What is the area of a parallelogram with a base of length b and height of length h,
and what line do I use for the height of the parallelogram?

What is A = bh, where h is the altitude of the parallelogram relative to the base.

500

What is the area of a triangle with a base of length b and height of length h,
and what line do I use for the height of the triangle?

What is A = 1/2 bh, where h is the altitude of the triangle relative to the base.

. . . and Perimeters

100

The perimeter of a polygon is?

What is the sum of the lengths of all sides of the polygon.

200

The area of a trapezoid with bases of lengths b₁ and b₂ and whose altitude has length h is given by?

What is A = 1/2h(b₁ + b₂)

300

The area of any quadrilateral with perpendicular diagonals of length d₁ and d₂ is given by?

What is A = 1/2(d₁ * d₂)

400

Name at least two polygons that we have studied that have perpendicular diagonals

What is a kite, a rhombus, and a square.

500

The ratio of the areas of two similar triangles A₁/A₂ is equal to?

What is the square of the ratio of the lengths of any two corresponding sides; that is

A₁/A₂ = (a₁/a₂)²

Regular Polygons

100

What is the name of the line that is the radius of the circumscribed circle around a regular polygon,
and what are the properties of this line?

What is a radius (joins the center of the regular polygon to a vertex).
It bisects the interior angle to which it is drawn.

200

What is the name of the line that is the radius of the inscribed circle inside a regular polygon,
and what are the properties of this line?

What is an apothem (a line segment from the center of the regular polygon perpendicular to a side of the polygon).
It bisects the side it is drawn to.

300

The area of a regular polygon whose apothem has length a and whose perimeter is P is given by?

What is A = 1/2 aP

400

Describe the proof of the Theorem that the area A of a regular polygon whose apothem has length a and whose perimeter is P is given by A = 1/2 aP.

What is divide the regular polygon into isosceles triangles whose sides are the radii of the polygon,
and whose height is an apothem of the polygon.
Each triangle has an area of 1/2 as, where s is the length of one of the sides of the polygon.
There would be n triangles, where n is the number of sides.
Adding up all of the areas of the triangles and factoring, you get A = 1/2 aP

500

What is the area of an equilateral triangle with sides of length s, and give a proof for the formula.

What is A = s²/4 * √3
Proof: draw an altitude. This divides the equilateral triangle into two 30:60:90 triangles.
The hypotenuse is length s, the short side is length s/2,
and the long side (which is the altitude of the original triangle) is of length s√3/2.
The area of any triangle is A = 1/2 bh, and in this case, b = s and h = s√3/2
A = 1/2 * s * s√3/2 = s²/4 * √3

Circles

100

What is the ratio of the circumference of a circle to the diameter of a circle?

What is pi.

200

Give two formulas for the circumference of a circle.

What is C = (pi)d where d is the length of the diameter of the circle,
and C = 2(pi)r where r is the length of the radius of the circle.

300

In a circle whose circumference is C, the length of an arc whose degree measure is m is given by?

What is l = m/360 * C

400

The area of a circle whose radius has length r is given by?

What is A = (pi)r²

500

Given a washer, where the radius of the outside edge is R and the radius of the inside edge is r,
what is the area of the surface of the washer?

What is A = (pi)R² - (pi)r²

and more Circles

100

A sector of a circle is defined as?

What is a region of a circle bounded by two radii of the circle and the arc intercepted by those two radii.

200

In a circle of radius r, find the perimeter and area of a sector whose arc has degree measure m.

What is P = 2r + m/360 (2(pi)r)
A = m/360 (pi)r²

300

A segment of a circle is defined as?

What is a region in a circle bounded by a chord and its minor (or major) arc.

400

Describe how to find the area of a segment of a circle.

What is A_segment = A_sector - A_triangle

500

Given a triangle with perimeter P and an inscribed circle of radius r, what is the area of the triangle?
How do you prove the formula?

What is A = 1/2 rP
Proof: The sides of the triangle will be tangents to the inscribed circle.
The radii of the circle will then be perpendicular to each side at the point of tangency.
Draw lines from the center of the inscribed circle to each vertex.
This forms three triangles whose bases are the sides of the triangle, and whose altitudes are radii of the circle.
The area of each smaller triangle is 1/2 rs_n
Adding the ares of each small triangle, and factoring, gives A = 1/2 rP