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

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

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Күн бұрын

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@AmirgabYT2185 3 ай бұрын

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.

@imetroangola17 3 ай бұрын

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 3 ай бұрын

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 3 ай бұрын

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

@imetroangola17 3 ай бұрын

@@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.

@imetroangola17 3 ай бұрын

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

@RAG981 3 ай бұрын

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

@dhorasokoli47 3 ай бұрын

In short, it could be solved like this: The height comes from the solution of the equation 13^2-x^2=14^2-(15^2-x^2)^2 because the height is expressed by the Pythagorean theorem in two forms in two right triangles. The solution gives x=6,6. Then apply T.P. in one of the triangles and the height h=11.2 comes out. In the end S=[(25+10)×11,2]÷2= 196cm^2

@TyyylerDurden 2 ай бұрын

Yeah, solved it the same way.

@baselinesweb 2 ай бұрын

I took the long way, but got that answer too. I made a regular trapezoid (13 on both left and right) and then the full right triangle had a base of x for 13 and x+x1 for 14. It was fun.

@srizzio 2 ай бұрын

Cool problem. Used different strategy. Created one triangle by taking the left and right sides of the trapezoid. Hence, a triangle with sides 13, 14, and 15. Used Heron's formula to calculate area of triangle. Took that value and calculated height of triangle, which is the height of the trapezoid. I dumped that value into the area formula for a trapezoid. Solution.

@kaliprasadguru1921 Ай бұрын

Draw BE parallel to AD. Then EC =15 . Determine area of triangle BEC . Then determine its height . Now determine the area of Trapezium.

@murdock5537 3 ай бұрын

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

@RealQinnMalloryu4 3 ай бұрын

(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).

@josephkt6830 Ай бұрын

Very good illustration.thanks

@kateknowles8055 Ай бұрын

Let height of trapezium be h. Draw AE and BF both equal to h , so that DEFC is a straight line. Area of triangle ADE = 0.5h x DE. Area of triangle BFC = 0.5 h x FC . Area of AEFB (the rectangle) is 10h. hxh= 13x13- DExDE= 14x14- FCxFC by Pythagors'theorem in both triangles FCxFC -DExDE = 14x14-13x13 = 196-169 =27 (FC+DE)(FC-DE)=27 FC+DE+10= 25 so FC-DE = 27/ 15 = 9/5 FC+DE=15 FC-DE=9/5 =1.8 2xFC = (15x5+9)/5 FC=8.4 DE= 6.6 13x13- 6.6x6.6= (13+6.6)(13-6.6)= 19.6 x 6.4= 196 x64/(10x10) = h x h. So h= 14x 8/10 = 112/10 = 11.2 Area of this trapezium = 11.2 (10 + 4.2 + 3.3) = 11.2 x 17.5 = 2.8 x 70 = 196 square units .

@wes9627 2 ай бұрын

f(h)=√(13^2-h^2)+√(14^2-h^2)-15=0 f'(h)=-h/√(13^2-h^2)-h/√(14^2-h^2) Newton-Raphsin iteration: Start with h=10 and iterate h←h-f(h)/f'(h) until h=11.200... The area is then (1/2)(10+25)11.2=196 sq units.

@santiagoarosam430 3 ай бұрын

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.

@satrajitghosh8162 3 ай бұрын

Draw AE || BC that cuts BC at E Area ( Trapezium ABCD) / area ( ∆ ADE) = ( AB + CD)/DE = ( AB + CD) / ( CD - AB) = ( 25 + 10)/( 25 - 10) = 7/3 Area ( ∆ ADE) of sides 13, 14, 15 = √ ( 21 * 8 * 7 * 6) sqr unit = 7 * 3 * 4 sqr unit = 84 sqr unit Area of Trapezium ABCD = (7/3) * 84 sqr unit = 196 sqr unit Corollary : Area of a trapizum with obleque sides a, b and, parallel sides x, y is (x> y) ∆ ( a, b, (x - y)) * ( x + y)/( x - y) Here ∆ ( a, b, c = (x - y)) = √ ( s * ( s - a) ( s - b)( s - c)) s = ( a + b + c)/2

@alster724 2 ай бұрын

Nice use of the Herons on Method 2

@subbugowda4204 Ай бұрын

2 parellal sides divided by 2 into other 2 sides divided by 2 and if multiplied how ?

@adgf1x 3 ай бұрын

ar.=196sq unit.ans

@shobavp5777 2 ай бұрын

AB is not parallel to CD because AD =13, BC=14 if AD=BC, AB and CD parallel

@marinaasstra2604 2 ай бұрын

По условию же трапеция

@shivamrathaur7195 2 ай бұрын

Bhai parallel hogi because AD and BC are transversal line not perpendicular distance between these parallel line.

@viswanathanlakshminarayana1576 2 ай бұрын

what abt nos 13, 17,, 19 , to divide by these nos

@adgf1x 22 күн бұрын

at of trapziu35×5.6=196. ans

@subbaraob8474 Ай бұрын

Super solution

@田中勉-c7c 3 ай бұрын

99/15は33/5と約分可能。

@daakudaddy5453 3 ай бұрын

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

@MathBooster 3 ай бұрын

I just didn't noticed it at that time.

@imetroangola17 3 ай бұрын

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

@sanils7260 2 ай бұрын

10+25/2=17.50, 13+14/2=13.50 17.5*13.50=236.25

@prossvay8744 3 ай бұрын

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

@quigonkenny 3 ай бұрын

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

@adgf1x 22 күн бұрын

alt. of trap=11.2 unit.

@robertloveless4938 3 ай бұрын

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

@juliocesargranafernandez2106 3 ай бұрын

Se complica con un desarrollo tan extenso y es más fácil que la tabla del uno.

@freel0077 26 күн бұрын

Simple is : Draw 2, 90^ triangles and one rectangle from this shape. Area of 90^ triangle= 1/2 ab Area of rectangle = ab ; a is length, b is height. Add all.

@sorourhashemi3249 3 ай бұрын

Thanks but so easy

@ramesankd1604 Ай бұрын

17.5

@Jaahquubel 2 ай бұрын

x = 99/15 = 33/5

@gnanadesikansenthilnathan6750 2 ай бұрын

Nice problem

@ShoaibAnwar-p7s 2 ай бұрын

17'- 6" × 13' - 6".

@wasimahmad-t6c 2 ай бұрын

189.4

@何柏希 3 ай бұрын

Why you guys not use S=

@Augustus9720 3 ай бұрын

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

@abdelmoniemzamli2446 2 ай бұрын

المساحة ١٩٦ وحدة مربعة

@wasimahmad-t6c Ай бұрын

10+25=35÷2=17.5×11.2=196(100%raite)

@spyadav2244 25 күн бұрын

Not right.

@adgf1x Ай бұрын

alt.11.2

@dhanrajchaudhari5627 3 ай бұрын

Ca is 25-x

@dhanrajchaudhari5627 3 ай бұрын

Sorry QC is 25-X

@venkatnagar27 Ай бұрын

Ans.210 sq.ft. Area

@wasimahmad-t6c 2 ай бұрын

195.6519

@soojin1 3 ай бұрын

Area= 196

@isaacaddae4207 2 ай бұрын

Please I won't JHs2 and 3

@surajrajanayake-eq3fy 3 ай бұрын

2nd Method 13.58min What is S

@vivificateurveridique1420 3 ай бұрын

he applied: To calculate the area of a triangle using its semi-perimeter, you can use Heron's formula. Here’s how it works: Calculate the semi-perimeter (s): 𝑠 = 𝑎 + 𝑏 + 𝑐 2 s= 2 a+b+c where 𝑎 a, 𝑏 b, and 𝑐 c are the lengths of the sides of the triangle. Calculate the area (A) using Heron's formula: 𝐴 = 𝑠 ( 𝑠 − 𝑎 ) ( 𝑠 − 𝑏 ) ( 𝑠 − 𝑐 ) A= s(s−a)(s−b)(s−c)