Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [triangles]

For questions about properties and applications of triangles.

A triangle is a polygon with three sides. It is an important geometric figure, because any polygon can be subdivided into triangles.

Triangles can be classified by the number of sides they have that have equal length

  • All three sides of an equilateral triangle have equal length.
  • An isosceles triangle has at least two sides of equal length.
  • A scalene triangle is a triangle that is not isosceles, that is, it has no sides with equal length.

A triangle may also be classified by describing its angles. A triangle is said to be a right triangle if it contains a right angle, and obtuse triangle if it contains an obtuse angle, or an acute triangle if all three of its angles are acute.

6703 questions
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Need help solving -

i was writing an paper on solutions of triangles when i encountered this sum - In a $\Delta$ ABC , P is an interior point such that $\angle PAB = 10^\circ$ , $\angle PBA = 20^\circ$ , $\angle PCA = 30^\circ$ , $\angle PAC = 40^\circ$ and a , b , c…
Sri Krishna
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The fastest way to find an obtuse triangle based on several sets of lengths

A question from ACT math: I'm wondering except using the law of cosine to get the answer K, is there any faster way to figure it out?
HypnoticBuggyWraithVirileBevy
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Find the sum of all possible values of a side of a triangle given a similar triangles side and area

Question: The difference in the areas of two similar triangles is 48 square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is 3…
user807252
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Finding Origin using 2 Points, Angle and Plane

I'm having a problem of finding the origin $O(x,y,z)$ of 2 3D points $P_1 (1,2,2)$ and $P_2 (2,3,5)$, distance $OP_1 = OP_2$, angle = 60 degrees and Plane Equation: $$E: 1.15714286x + 1.8547619y-z-2.86101191=0.$$ I used the distance equation, plane…
Samann Long
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Finding $\frac{c\sin(A-B)}{a^2-b^2}-\frac{b\sin(C-A)}{c^2-a^2}$

If $a, b$ and $c$ (all distinct) are the sides of a triangle ABC opposite to the angles $A, B$ and $C$, respectively, then $\frac{c\sin(A-B)}{a^2-b^2}-\frac{b\sin(C-A)}{c^2-a^2}$ is equal to $?$ By opening, $\sin(A-B)$ as $\sin A\cos B-\cos A\sin…
aarbee
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In $\Delta ABC$ prove that $\frac{a-b\cos C}{b-a\cos C}=\frac{\cos B}{\cos A}$

LHS: $$\frac{a-b(\frac{a^2+b^2-c^2}{2ab})}{b-a(\frac{a^2+b^2-c^2}{2ab})}$$ $$\frac{a^2+c^2-b^2}{b^2+c^2-a^2}$$ I can’t solve further. You can’t devide the numerator and denominate by $2ac$ or $2bc$, otherwise it would have been really easy
Aditya
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Counting Heronian triangles based on area to perimeter ratios

For Heronian triangles t with the perimeter p and area a, there are five distinct triangles with the area equal to the perimeter: If a=p, count(t) = 5, if a=2p, count(t)=18. For the area being an integer multiple of the perimeter, what is the…
Jamie M
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In the triangle ABC, the altitudes are in Harmonic progression then-

$$\frac{2}{\frac{2\Delta}{b}}=\frac{1}{\frac{2\Delta}{a}}+\frac{1}{\frac{2\Delta}{c}}$$ $$2b=a+c$$ Thus a,b and c are in an arithmetic progression. But the answer is $\sin A,\sin B,\sin c$ are in AP. I understand that both are equivalent, but just…
Aditya
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Find the value of $(r_1-r)(r_2-r_3)$

$$=(\frac {\Delta}{s-a}-\frac{\Delta}{s})(\frac{\Delta}{s-b}-\frac{\Delta}{s-c})$$ $$=\Delta ^2(a)(b-c)\frac {1}{\Delta ^2}$$ $$a(b-c)$$ But the answer given is $a^2$ $r_1,r_2, r_3$ are the ex radii on sides a, b and c respectively.
Aditya
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Geometry: given a fixed area and vertex angle, prove that an isosceles triangle would minimize the length of the base

I was working on this problem and could intuitively see how this is true. I tried to solve it on my own, but the only "proof" I could come up with was that because the non-base sides increase at different rates in order to keep the area the same,…
Bluedragon01313
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If two sides of a triangle are given, how many triangles are possible?

I thought myself that you could rotate one arbitrary side, which is connected to the other side, over 90 degrees to the left and 90 degrees to the right, so basically there are infinitely many triangles possible. Am I right?
mathomato
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How to find the length of the sides of an isosceles right triangle given its hypotenuse ? (Without trigonometry)

I was doing my math homework when this question came. There's an isosceles triangle with its base length which equals 13, and the other sides' length n. The question is, "if this triangle was a right triangle, what's the value of n ?". PS. I'm a 9th…
PrograMed IP
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How to find angle of triangle inside circle?

Hi there, In the above diagram: Q is the center of the circle, PAT is a tangent to the circle, PR is parallel to AC, angle CAT = x. Prove that angle ABC = x. I started off by going 90 - x to find CAB. Then used co-interior angles to find AQR, and…
chris
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Finding expressions using graphs

Here is a question Anyone know how to solve this or any hints to get me started as I am stuck. The diagram shows a triangle ABC with sides BC a and AC b The points D,E and F lie on the sides AC,AB and BC, respectively, so that CDEF is a…
Alex
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Geometrical Proof of $\sin A+\sin B+\sin C= 4\cos(A/2)\cos(B/2)\cos(C/2)$

I have been seeking for geometrical proof of $\sin A+\sin B+\sin C= 4\cos(A/2)\cos(B/2)\cos(C/2)$ Where $A, B, C$ are angles of a triangle. Do you have any ideas about it?
Soling
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