John L.

asked • 05/24/17# Given: LMNB is a square, LM = 20cm, P ∈ LM, K ∈ PN, PK= 1/5 PN, LP = 4 cm; Find: Area of LPKB.

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## 1 Expert Answer

Christine L. answered • 05/25/17

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UCB Grad & HS Teacher: HS/College Subjects, Math, Test Prep, etc

To start, note that PMN is a right triangle. Draw a line perpendicular through point K that is perpendicular to PM and label the intersection as point Q. Observe that angle PQK is a right angle and that both triangles PKQ and PNM contain angle MPN. By AA, PKQ ~ (is similar to) PNM. This allows us to say something about the ratio of their side lengths.

Since the two triangles are similar, the following holds PK/PN = PQ/PM=KQ/MN. Since we were given that PK = (1/5)PN, we have the constant of proportionality that allows us to find the missing values.

KQ = (1/5)MN = (1/5)*20 = 4.

PQ = (1/5)PM = (1/5)*16 = 16/5.

In your diagram, make sure to show that PQ = 16/5 and QK = 4.

Now, draw a line through point K that is perpendicular to LB and label the intersection as point R. Then, draw a line through point P that is perpendicular to RK and label the intersection as point T. NOTE: We have now shown that LPKB is a composition of 3 figures: LPKB = LPTR + PTK + KRB.

Observe that since PQKT forms a rectangle so PT = QK = 4 and PQ = TK = 16/5.

Since LP = PT = 4, LPTR is a square and has an area of 16.

PTK is a right triangle so its area is ½ * PT * TK = ½ * 4 * 16/5 = 32/5.

BRK is a right triangle so its area is ½ * BR * RK = ½ * BR * (RT + TK). Since LPTR is a square, LR = RT = 4. Since LMNB is a square, LB = 20. Since LB = LR + RB, we know 20 = 4 + RB and thus RB = 16.

This means that the area of triangle BRK = ½ * BR * (RT + TK) = ½ * 16 * (4 + 16/5) = ½ * 16 * (36/5) = 288/5.

Thus, the area of LPKB = the summed areas of LPTR + PKT + BRK = 16 + 32/5 + 288/5 = 80.

So, the area is 80 cm^2.

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Mark M.

05/24/17