This presentation is the property of its rightful owner.
1 / 7

# EXAMPLE 4 PowerPoint PPT Presentation

Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M. SOLUTION.

EXAMPLE 4

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

#### Presentation Transcript

Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M.

SOLUTION

Place PQOwith the right angle at the origin. Let the length of the legs be k. Then the vertices are located at P(0, k), Q(k, 0), and O(0, 0).

EXAMPLE 4

Apply variable coordinates

=

=

=

= k

2

k

k

2

2

=

0 + k , k +0

M( )

M( , )

2

2

2

2

2

2

2

2

2

k + (– k)

2k

k + k

(k–0) + (0–k)

EXAMPLE 4

Apply variable coordinates

Use the Distance Formula to find PQ.

PQ =

Use the Midpoint Formula to find the midpoint Mof the hypotenuse.

Write a coordinate proof of the Midsegment Theorem for one midsegment.

E(q+p, r)

D(q, r)

=

=

GIVEN :

PROVE :

SOLUTION

1

DE is a midsegment of OBC.

2

DE OCand DE = OC

STEP1

Place OBCand assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of Dand E.

2q + 2p, 2r + 0

2q + 0, 2r + 0

E( )

D( )

2

2

2

2

EXAMPLE 5

Prove the Midsegment Theorem

STEP 2

Prove DE OC. The y-coordinates of Dand Eare the same, so DEhas a slope of 0. OCis on the x-axis, so its slope is 0.

1

Because their slopes are the same, DE OC.

2

Prove DE = OC. Use the Ruler Postulate

to find DEand OC.

STEP3

= 2p

2p – 0

(q +p) – q

OC=

= p

DE=

So, the length of DEis half the length of OC

EXAMPLE 5

Prove the Midsegment Theorem

In Example 5, find the coordinates of F, the midpoint of OC. Then show that EF OB.

SOLUTION

Given:FE is a midsegment.

Prove:FE OB

The midpoints are E (q + p, r ) and F = F (p, 0). The slope of both FEand OBis so

EF ||OB

r

q

1

Also, FE = q2 + r2and OB = 2 q2 + r2, so FE = OB.

2

for Examples 4 and 5

GUIDED PRACTICE

7.

Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJa right triangle? Find the side lengths and the coordinates of the midpoint of each side.

8.

J(m,0)

H(m,n)

STEP1

Place OHJwith the right

angle at the origin

The vertices are

O(0,0)

, ,

( 0,0)

(m,n)

(m,0)

O

H

J

for Examples 4 and 5

GUIDED PRACTICE

SOLUTION

=

=

=

HJ =

OH =

= n

m

2

0 + m , n +0

n

C( )

C( , )

=

2

2

2

2

2

2

2

2

2

2

2

2

2

0 + (– n)

m + (– n)

Yes, OHJa right triangle

m + n

(0 – m) + (0 –n)

(m – m) + (0 –n)

for Examples 4 and 5

GUIDED PRACTICE

STEP2

Use the distance formula to find OH and HJ

Use the Midpoint Formula to find the midpoint Cof the hypotenuse.