Using the Distance and Midpoint Formulas in Proofs

A circle with radius $a$ intercepts the $x -$ and $y -$ axis once each. It also intersects a circle with radius $r$ and center at the origin at exactly one point.

Two circles. One with radius a that intercepts the x- and y-axis once each. One with radius r and center at the origin. The circles intersect each other at exactly one point.

Write an expression for the radius of the smaller circle in terms of

a .

Answer in exact form.

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Segment	Points	Substitute	Simplify
$\overline{A C}$	$A (- 3, 0)$ $&$ $C (4, - 1)$	$A C =$ $(4 - (- 3))^{2} + (- 1 - 0)^{2}$	$A C =$ $52$
$\overline{A B}$	$A (- 3, 0)$ $&$ $B (- 2, 2)$	$A B =$ $(- 2 - (- 3))^{2} + (2 - 0)^{2}$	$A B =$ $5$
$\overline{B C}$	$B (- 2, 2)$ $&$ $C (4, - 1)$	$B C =$ $(4 - (- 2))^{2} + (- 1 - 2)^{2}$	$B C =$ $35$

$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Diagonal	Endpoints	Substitute	Simplify
$\overline{B D}$	$B (- 2, 2)$ $&$ $D (3, - 3)$	$M (\frac{- 2 + 3}{2}, \frac{2 + ( - 3 )}{2})$	$M (0.5, - 0.5)$ $✓$
$\overline{A C}$	$A (- 3, 0)$ $&$ $D (4, - 1)$	$M (\frac{- 3 + 4}{2}, \frac{0 + ( - 1 )}{2})$	$M (0.5, - 0.5)$ $✓$

$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Side	Endpoints	Substitute	Simplify
$\overline{A B}$	$A (0, 0)$ and $B (2, 2.5)$	$M_{1} (\frac{0 + 2}{2}, \frac{0 + 2 . 5}{2})$	$M_{1} (1, 1.25)$
$\overline{B C}$	$B (2, 2.5)$ and $C (6, 0)$	$M_{2} (\frac{2 + 6}{2}, \frac{2 . 5 + 0}{2})$	$M_{2} (4, 1.25)$

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Segment	Endpoints	Substitute	Simplify
$\overline{M_{1} M_{2}}$	$M_{1} (1, 1.25)$ and $M_{2} (4, 1.25)$	$M_{1} M_{2} = (4 - 1)^{2} + (1.25 - 1.25)^{2}$	$M_{1} M_{2} = 3$
$\overline{B C}$	$A (0, 0)$ and $C (6, 0)$	$A C = (6 - 0)^{2} + (0 - 0)^{2}$	$A C = 6$

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Segment	Points	Substitute	Simplify
$\overline{A D}$	$A (0, 0)$ $&$ $D (3, 0)$	$A D =$ $(3 - 0)^{2} + (0 - 0)^{2}$	$A D =$ $3$
$\overline{B C}$	$B (2, 2)$ $&$ $C (5, 2)$	$B C =$ $(5 - 2)^{2} + (2 - 2)^{2}$	$B C =$ $3$

Using the Distance and Midpoint Formulas in Proofs

Catch-Up and Review

Investigating the Properties of a Parallelogram

Distance Formula

Proof

Practice Finding the Distance Between Two Coordinate Pairs

Classifying a Parallelogram Using Its Coordinates

Hint

Solution

Determining If a Point Is On a Circle's Circumference

Hint

Solution

Definition of a Midpoint

Midpoint

Midpoint Formula

Proof

Practice Finding Midpoints Using the Midpoint Formula

Proving the Properties of a Parallelogram

Answer

Hint

Solution

Using Rigid Motions to Prove a Theorem

Extra

Proving the Triangle Midsegment Theorem by Rigid Motions

Answer

Hint

Solution

Proving the Properties of a Parallelogram

Answer

Hint

Solution

Using the Distance and Midpoint Formulas in Proofs

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Triangle Midsegment Theorem
The segment that connects the midpoints of two sides of a triangle — a midsegment — is parallel to the third side of the triangle and half its length.