Deck 10: Analytical Geometry

Question 1

Let a and b represent positive real numbers. Use the right triangle with vertices at R(0,0), S(2a,0), and T(0,2b) to prove the following theorem.
"The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the
triangle."

Answer 1

With being horizontal and being vertical, the right angle is at vertex R. Where M is the midpoint of the hypotenuse , we have so that Applying the Distance Formula, we see that the distances from M to the vertices are , ,
and . That is, and it follows that midpoint of hypotenuse is equidistant from vertices R, S, and T.

Question 2

Let a and b represent positive real numbers. Explain why the triangle with vertices at A(a,0), B(0,a) and C(0,0) is an isosceles right triangle.

Answer 2

For , we apply the Slope Formula to show that or 0; thus, is horizontal. Also, , which is undefined; thus, is vertical. Then ,
so is a right triangle.
Because is horizontal, its length is the positive difference is x coordinates; that is, or a. Because is vertical, its length is the positive difference in its y coordinates; that is, or a. Then and is also isosceles.

Question 3

Let a, b, and c represent positive real numbers. Given the rhombus ABCD with vertices at A(0,0), B(a,0), C(a+b,c) and D(b,c), prove the following theorem.
"The diagonals of a rhombus are perpendicular."

Answer 3

With vertices as stated, ABCD appears to be parallelogram. For ABCD to further represent a rhombus, and must be congruent. But only if (the result of applying the Distance Formula); in turn, .
Having established the conditions for the rhombus, we will now use the Slope Formula to show that the diagonals are perpendicular. so . Similarly, so .
Then the product of the slopes of these diagonals is , so .
By substitution (recall that ), .
With , it follows that .

Question 4

Let a, b, and c represent positive real numbers. Consider the parallelogram ABCD with vertices at A(0,0), B(a,0), C(a+b, c), and D(b,c). In order that ABCD further represents a rhombus, prove that .
[Note: No drawing provided.]

Question 5

Let a, b, and c represent positive real numbers. Use the drawing in which the vertices of are A(0,0), B(2a,0), and C(2b,2c) to prove the following theorem.
"The line segment determined by the midpoints of two sides of a triangle is parallel to the third side of the triangle."

Question 6

Let a, b, and c represent positive real numbers. Use the figure in which has vertices at A(0,0), B(a,0), C(a+b,c), and D(b,c) to prove the following theorem.
"If the diagonals of a parallelogram are equal in length, then the parallelogram is a rectangle."

Question 7

Let a, b, and c represent positive real numbers. Given that quadrilateral ABCD has vertices at A(0,0), B(a,0), C(a+b,c), and D(b,c), explain why ABCD must be a parallelogram.

Question 8

Let a and b represent positive real numbers. Use the drawing in which the vertices of rectangle RSTV are R(0,0), S(2a,0), T(2a,2b), and V(0,2b) to prove the following theorem.
"The diagonals of a rectangle bisect each other."