A Very Nice Geometry Problem | Find the area of the trapezium | 2 Methods

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[@imetroangola4943]

In method 2, it would be more interesting if you had found the area of the parallelogram and, after adding it to the area of the triangle, finally, we would have the area of the trapezoid: [ABPD]=h×BP=56/5 × 10=112, Therefore, [ABCD]=[ABPD]+[BCP] *[ABCD]=84+112=196*

[@quigonkenny]

Seems to me like you're just adding additional unnecessary steps. You use the area of the triangle to get the value of h, but once you have that it's unnecessary to find the area of the parallelogram, since you can use h to directly find the area of the whole trapezoid.

[@purnajinananandaavadhuta8605]

We need not find the height in the 2nd method. Area of the parallelogram = 84*10*2/15=112. Total area= 84+112=196

[@imetroangola4943]

​@@quigonkennyUnfortunately, you didn't understand what I meant, in the second method it would be more interesting not to use the formula for the area of the trapezoid, as it is easier to find the area of the parallelogram.

[@imetroangola4943]

​@@purnajinananandaavadhuta8605It can be done like this, but for the sake of explanation, the interesting thing is to find the height!

[@RAG981]

@@quigonkenny Area parallelogram is just 10x56/5 =112, and add to 84. That is much simpler.

[@AmirgabYT2185]

Let's drop two heights from the upper vertexes down to the larger base. They will be h each. Let segments we got by dropping heights be a (the left one) and b (the right one) and middle which is equal to the smaller base and that is 10. Then we got: h²=13²-a² On the other hand: h²=14²-b² So: 169-a²=196-b². And also: a+b+10=25 After few changes, we can make a system: b²-a²=27 a+b=15 Let's solve it. (b-a)(b+a)=27 a+b=15 15(b-a)=27 b-a=1,8 We got a new easier system: b-a=1,8 a+b=15 Let's add equations to each other: 2b=16,8 b=8,4=42/5 a=6,6=33/5 Let's find h: h²=196-(42/5)² 196-1764/25=h² 3136/25=h² h is a height so h>0 => h=56/5=11,2 S=(10+25)11,2/2=35•5,6=196 Answer: 196 square units.

[@santiagoarosam430]

DC=a+10+(25-10-a)→ 13²-a² =h²= 14²-(15-a)²→ a=33/5→ h²=13²-(33/5)²→ h=56/5→ Área ABCD =[(10+25)/2]*(56/5)=196 ud². Gracias y saludos.

[@robertloveless4938]

I imagined method 2 right away. BUT, somewhere, I went astray and got the answer wrong. Oh well.

[@quigonkenny]

Draw AE, where E is the point on DC where AE amd BC are parallel. This creates parallelogram ABCE, with sides of 14 and 10, and triangle ∆AED with sides of 13, 14, and 25-10 = 15. Let ∠EDA = θ. By the law of cosines: cos(θ) = (AD²+ED²-AE²)/2(AD)ED cos(θ) = (13²+15²-14²)/2(13)(15) cos(θ) = (169+225-196)/390 cos(θ) = 198/390 = 33/65 sin²(θ) = 1 - cos²(θ) = 1 - (33/65)² sin²(θ) = 1- 1089/4225 = 3136/4225 sin(θ) = √(3136/4225) = 56/65 (1/2)(ED)h = (1/2)AD(ED)sin(θ) h = 13(56/65) = 56/5 Trapezoid ABCD: [ABCD] = h(a+b)/2 = (56/5)(10+25)/2 [ABCD] = (56/5)(35)/2 = 28(7) = 196 sq units

[@murdock5537]

x = √(13^2 - h^2) = 15 - √(14^2 - h^2) → h = 56/5 → area ABCD = 125h or: ∆ BPC → CP = 15; BP = 13; BC = 14; BPC = θ → cos⁡(θ) = 33/65 → sin⁡(θ) = 56/65 = h/13 → h = 56/5 → area ABCD = 125h

[@prossvay8744]

Trapezoid area=1/2(10+25)(56/5)=196

[@田中勉-c7c]

99/15は33/5と約分可能。

[@RealQinnMalloryu4]

(13)^2=169 (14)^2=196 {169+196}=365 (10)^2=100 (25)^2=625{100+625}=725 {365+725}=1090/180°ABCD =6 .50ABCD 6.5^10 6.5^2^5 3^2.1^2^1 3^1.2^1 3.2 (ABCD ➖ 3ABCD+2).

[@sorourhashemi3249]

Thanks but so easy

[@soojin1]

Area= 196

[@rgcriu2530]

👍👍👍👍

[@何柏希]

Why you guys not use S=

[@Augustus9720]

S mean surface, A means area but they are actually the same just because the difference of habit.习惯不同而已

[@daakudaddy5453]

99/15 = 33/5 Would have made calculatioj much simpler. I don't know why you chose not to simplify it further...

[@MathBooster]

I just didn't noticed it at that time.

[@imetroangola4943]

​@@MathBoostero importante é que você resolveu! Também, é importante mostrar outras soluções! Matemática, sempre tem outras formas de chegar ao resultado.

[@SatyanarayanaMudunuri]

what us yesquaire? Awful pronunciation and explanation

[@surajrajanayake-eq3fy]

2nd Method 13.58min What is S