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."

Accepted Answer

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.

Accepted Answer

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."

Accepted Answer

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.]

Accepted Answer

The answer of 
-Let a, b, and c represent positive...

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."

Accepted Answer

The answer of 
-Let a, b, and c represent positive...

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."

Accepted Answer

The answer of 
-Let a, b, and c represent positive...

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.

Accepted Answer

The answer of 
-Let a, b, and c represent positive...

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."

Accepted Answer

The answer of 
-Let a and b represent positive real...

Quiz 10: Analytical Geometry

-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."

-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.

-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."

-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.]

-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."

-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."

-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.

-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."