Improve Your Geometry Skills with Practice Problems

The most general value of x satisfying the equations $\tan <!--\; \; -->x=-<!--\; -\; -->1$ and $\cos <!--\; \; -->x=1/\sqrt{2}$, is found to be $x=2n\mathrm{\pi <!--\; \pi \; -->}+\frac{7\mathrm{\pi <!--\; \pi \; -->}}{4}$

Given equation:
$\frac{\sin <!--\; \; -->(x)+\sin <!--\; \; -->(5x)-<!--\; -\; -->\sin <!--\; \; -->(3x)}{\cos <!--\; \; -->(x)+\cos <!--\; \; -->(5x)-<!--\; -\; -->\cos <!--\; \; -->(3x)}=\tan <!--\; \; -->(3x),$
what is the easiest way to solve it?

I have been recently studying a C.G. Hempel's article on mathematical truth and pointed out his following quotation: "Every concept of mathematics can be defined by means of Peano's three primitives,and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms".

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)

Prove $|x+y{|}^2+|x-<!--\; -\; -->y{|}^2=2|x{|}^2+2|y{|}^2$ if $x,y\in <!--\; \in \; -->{\mathbb{R}}^k$.
Interpret this geometrically, as a statement about parallelograms.
I've shown that the expression given equates to $x\cdot <!--\; \cdot \; -->x+2x\cdot <!--\; \cdot \; -->y+y\cdot <!--\; \cdot \; -->y+x\cdot <!--\; \cdot \; -->x-<!--\; -\; -->2x\cdot <!--\; \cdot \; -->y+y\cdot <!--\; \cdot \; -->y$. But what does this have to do with parallelograms?

If the radius of an inscribed circle in a right triangle is 3 cm and the non-hypotenuse side is 14 cm, calculate triangle's area. I've tried to use the formula: $r=\frac{a+b-<!--\; -\; -->c}{2}$ but that doesn't help me that much.

The internal bisectors of the angles of a triangle ABC meet the sides in D,E,and F.Show that area of the triangle DEF is equal to $\frac{2\mathrm{\Delta <!--\; \Delta \; -->}\times <!--\; \times \; -->abc}{(b+c)(c+a)(a+b)}$,here $\mathrm{\Delta <!--\; \Delta \; -->}$ is area of triangle ABC

If I choose B as origin,C as(a,0),a is side length BC.what should i take coordinates of A,can i get answer with this approach or any better approach?

If ($x,y,z$) be the lengths of perpendiculars from any interior point P of a triangle $ABC$ on sides $BC$,$CA$ and $AB$ respectively then find the minimum value of :
$x^2+y^2+z^2$
The sides of triangle being $a,b,c$.

I thought of using Lagrange's method of multipliers but am not able to find another function in terms of $x,y,z$ and $a,b,c$Any help will be appreciated. Thanks.

Finding the slopes of internal or external angle bisectors, when given the slopes of the sides of a triangle, it's a simple question, the one you can easily find in any textbook of analytic geometry.

But the reverse problem : finding the slopes of the sides of a triangle, when only given the slopes of internal angle bisectors. Is it possible? Is there a straightforward formula for that?

How would I solve this trigonometric equation? $3\cos <!--\; \; -->x\cot <!--\; \; -->x+\sin <!--\; \; -->2x=0$
I got to this stage:
$3\cos <!--\; \; -->x=-<!--\; -\; -->2{\sin }^2<!--\; \; -->x$
Is is a dead end or is there a easier way solve this equation?

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From this point the surveyor moves 21.6 metres directly towards the tower and repeats the measurement. The angle of elevation of the top of the tower from this point is 68.4 degrees. Find the height of the tower.

The way I view Euclid's postulates are as follows:

A line segment can be made between any two points on surface A.

A line segment can be continued in its direction infinitely on surface A.

Any line segment can form the diameter of a circle on surface A.

The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.

If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.

The way I view the five postulates is simple. Each postulate defines some quality a surface has.

For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).

I want to determine what "quality" 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be "categorized" under some combination of postulates, but that is irrelevant.

I am merely asking:

What two surfaces individually violate the first postulate and violate the fourth postulate?

Take the Viviani curve intersection of a sphere and a cylinder. Analytically explain what happens to the intersection curve if you keep the radius of the sphere constant, fix one side of the cylinder at the origin, and change the radius of the cylinder. Especially, discuss the cases when the radius of the cylinder goes to 0 and $\mathrm{\infty <!--\; \infty \; -->}$

$\sin <!--\; \; -->x\cdot <!--\; \cdot \; -->\sin <!--\; \; -->2x\cdot <!--\; \cdot \; -->\sin <!--\; \; -->3x=\frac{\sin <!--\; \; -->4x}{4}$ - solving a trigonometric equation

Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles, using coordinate geometry.

First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the bisector of $a_1$ and $a_2$ which has minimal distance from $a_1$ and $a_2$ .
Second, if B denotes the degree 2 finite Blaschke product with zeros $a_1$ and $a_2$ , is the critical point of B which is in the disk equal to the midpoint of $a_1$ and $a_2$ ?

Focuses of the hyperbola $F_1(4,2),F_2(-<!--\; -\; -->1,-<!--\; -\; -->10)$ and the tangent equation $3x+4y-<!--\; -\; -->5=0$ are given. Find the semiaxes of the hyperbola.
I’ve tried using the definition of hyperbola and the equation for the tangent line to the quadratic curve to form some sort of system of equations, but it didn’t work out.

How can we use vectors and dot products to show that the diagonals of a parallelogram intersect at ${90}^{\circ <!--\; \circ \; -->}$ if and only if the figure is a rhombus?
I did the proof, but I realized my final answer would be a rectangle. (I know a rhombus is a type of rectangle, too). But I only want to prove the two diagonals are orthogonal.

Determine a so that the intersection line between the planes
$P_1:2x+ay-<!--\; -\; -->z=3$
$P_2:x-<!--\; -\; -->2y+az=5$
are parallel to the plane $P_3:2x+y+z=2$