example 2
DESCRIPTION
Use the following method to find the distance across a river, from point N to point P. Place a stake at K on the. near side so that NK NP. Find M , the midpoint of NK. Locate the point L so that NK KL and L , P , and M. are collinear. EXAMPLE 2. - PowerPoint PPT PresentationTRANSCRIPT
EXAMPLE 2 Use congruent triangles for measurement
Surveying
Use the following method to find the distance across a river, from point N to point P.
• Place a stake at K on thenear side so that NK NP
• Find M, the midpoint of NK .
• Locate the point L so that NK KL and L, P, and Mare collinear.
EXAMPLE 2 Use congruent triangles for measurement
• Explain how this plan allows you to find the distance.
SOLUTION
Because NK NP and NK KL , N and K are congruent right angles.
Then, because corresponding parts of congruent triangles are congruent, KL NP . So, you can find the distance NP across the river by measuring KL .
MLK MPN by the ASA Congruence Postulate.
Because M is the midpoint of NK , NM KM . The vertical angles KML and NMP are congruent. So,
EXAMPLE 3 Plan a proof involving pairs of triangles
Use the given information to write a plan for proof.
SOLUTION
GIVEN 1 2, 3 4
PROVE BCD DCE
In BCE and DCE, you know 1 2 and CE CE . If you can show that CB CD , you can use the SAS Congruence Postulate.
EXAMPLE 3 Plan a proof involving pairs of triangles
CBA CDA. You are given 1 2 and 3 4. CA CA by the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBA CDA.
To prove that CB CD , you can first prove that
Plan for ProofUse the ASA Congruence Postulate to prove that CBA CDA. Then state that CB CD . Use the SAS Congruence Postulate to prove that BCE DCE.
GUIDED PRACTICE for Examples 2 and 3
2. In Example 2, does it matter how far from point N you place a stake at point K ? Explain.
SOLUTION
No, it does not matter how far from point N you place a stake at point K . Because M is the midpoint of NK
GivenNM MK
Definition of right triangle
MNP MKL areboth right triangles
Vertical angleKLM NMP
ASA congruence MKL MNP
GUIDED PRACTICE for Examples 2 and 3
No matter how far apart the strikes at K and M are placed the triangles will be congruent by ASA.
3. Using the information in the diagram at the right, write a plan to prove that PTU UQP.
GUIDED PRACTICE for Examples 2 and 3
Given TU PQ
Given PT QU
Reflexive property PU PU
This can be done by showing right triangles QSP and TRU are congruent by HL leading to right triangles USQ and PRT being congruent by HL which gives you PT UQ
STATEMENTS REASONS
SSSPTU UQP
PTU UQP By SSS