1. The base of the pyramid ABCD is the isosceles triangle ABC (AB= 40 and AC= BC=25). The lateral edge CD=8 is perpendicular to the base. What is the total surface area of the pyramid?
2. The base of the pyramid ABCD is the triangle ABC (AB= 13, BC= 14, AC= 15). The lateral edge AD=16 opposite the medium-sized side of the base is perpendicular to the base. What is the total surface area (S1) of the pyramid?

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1 Answer

(1) In ∆ABC, join C to midpoint E of AB. CE is the perpendicular bisector of AB because ∆ABC is an isosceles triangle. BE=AE=20. CE=√(BC²-BE²)=√(625-400)=15. ∆s BCD and ACD are right triangles because CD is perpendicular to the base triangle ABC. DE is the hypotenuse of ∆CDE, so DE=√(CD²+CE²)=√(64+225)=17. DE is the height of ∆ABD, so the area of ∆ABD=½(AB)(DE)=½×40×17=340. Areas of ∆s BCD and ACD are the same so their combined area is 8×25=200. The area of ∆ABC=½×40×15=300.

The total surface area of the pyramid is 340+200+300=840 square units.

(2) ∆s ACD and ABD are two right triangles. We can find the squares of the lengths of their hypotenuses:

CD²=AD²+AC²=256+225=481,

BD²=AD²+AB²=256+169=425.

If DE is the perpendicular from D on to BC, then we have two right triangles, BDE and CDE. DE²=BD²-BE²=CD²-CE². But CE=BC-BE=14-BE.

BD²-CD²=BE²-(14-BE)²=(BE+14-BE)(BE-14+BE)=14(2BE-14).

425-481=28BE-196, BE=(425+196-481)/28=5 and CE=9.

Therefore, DE²=425-25=481-81=400, DE=20, which is the height of ∆BCD.

If AF is the perpendicular from A on to BC, we have two right triangles, ABF and ACF. AF²=AB²-BF²=AC²-CF², and BF+CF=BC=14.

AB²-AC²=BF²-(14-BF)²=14(2BF-14),

169-225+196=28BF, BF=5 (so E and F are the same point), CF=CE=9.

AF²=169-25=225-81=144, AF=12, height of ABC with base BC=14.

Area of ∆ABC=½×14×12=84.

Area of ∆ABD=½×13×16=104.

Area of ∆ACD=½×15×16=120.

Area of ∆BCD=½×14×20=140.

S1=84+104+120+140=448 square units.

by Top Rated User (825k points)

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