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7. In the diagram, LMN = ONM= 90°. P is the midpoint of MN,
MN = 2ML and MN = NO. Prove that:
a) the triangles MNL and NOP are congruent
b) OÊN = LÑO
c) LỘO=90°

In the diagram,LMN = ONM= 90°. P is the midpoint of MN, MN = 2ML and

MN = NO

Step-by-step explanation:

Step 1

(A)  Angle LMN = angle ONM [given]

MN = NO [given]

P is the midpoint of MN [given]

MN = 2 PN [definition of midpoint]

MN = 2 ML [given]

2 ML = 2 PN [transitivity of equality]

ML = PN [division property of equality]

Triangles MNL and NOP are congruent [side-angle-side]

Step 2

(B) Angles LMN and ONM measure 90° [given]

Measures of angles LMN, MNL, and NLM sum to 180° [angles of a triangle]

Measures of angles MNL and NLM sum to 90° [subtraction property of equality]

Measures of angles MNL and LNO sum to 90° [adjacent angles]

Measures of angles MNL and NLM = measures of angles MNL and LNO [transitive property of equality]

Measures of angles NLM and LNO are equal [subtraction property of equality]

Angles NLM and LNO are congruent [definition of congruent angles]

Triangles MNL and NOP are congruent [proved in part (a)]

Angles OPN and NLM are congruent [corresponding angles of congruent triangles]

Angles OPN and LNO are congruent [transitive property of angle congruence]

(C)  Angles QNO and LNO are congruent [same angle]

Angles NOQ and NOP are congruent [same angle]

Angles OPN and LNO are congruent [proved in part (b)]

Angles QNO and OPN are congruent [transitive property of angle congruence]

Triangles NOP and QON are similar [angle-angle similarity]

Angles NQO and PNO have the same measure [corresponding angles of similar triangles]

Angle MNO measures 90° [given]

Angle PNO measures 90° [same angle]

Angle NQO measures 90° [equal angles]

LN is perpendicular to PO [definition of perpendicular]

Angle LQO measures 90° [definition of perpendicular]

Thus we can say that LQO=90°

sasasadaassasasasa opk ok you know

answer is  

a.

5.2 inches

hope you get this right.